Phonetic Trajectory Memory (PTM)

• Phonetic Trajectory Memory (PTM) is a novel paradigm that maps language onto a 16-dimensional ergodic torus, enabling continuous, infinite-context memory via geometric trajectories.
• It applies a three-step process—acoustic injection, ergodic evolution, and manifold resonance—to achieve constant-time retrieval and over 3000× compression relative to traditional key-value caches.
• The architecture blends geometric and semantic likelihoods to suppress hallucinations and maintain retrieval accuracy up to 92%, offering robust performance even with long token streams.

An ergodic phonetic manifold is a compact, high-dimensional topological space equipped with an ergodic dynamical process onto which phonetic information is continuously mapped, evolved, and retrieved. This concept underlies the Phonetic Trajectory Memory (PTM) architecture, which reinterprets the memory of sequential data—such as language—in terms of persistent trajectories on a geometric structure governed by irrational rotations, as opposed to traditional finite or growing key-value (KV) caches. PTM enables unprecedented compression, retrieval fidelity, and constant-time access, establishing a new paradigm for a theoretically infinite context memory in LLMs (Houichime et al., 23 Dec 2025).

1. Topology and Dynamical Properties of the Ergodic Phonetic Manifold

The core state-space is a 16-dimensional torus, $\mathbb{T}^{16} = \mathbb{R}^{16}/\mathbb{Z}^{16}$, constructed by identifying the opposite faces of the unit hypercube. The torus has unit volume ($V=1$), and distances are measured using the Lee (toroidal) metric:

$$d_{\mathbb{T}}(u, v) = \sqrt{\sum_{i=1}^{16} \min\bigl(|u_i - v_i|,\;1 - |u_i - v_i|\bigr)^2}.$$

Temporal evolution is achieved through a rotation operator, $\mathcal{R}(t) \in SO(16)$, implemented as a block-diagonal of eight $2\times 2$ planar rotors, each associated with an angular frequency $\omega_k = \pi\sqrt{p_k}$ ($p_k$ is the $k$-th prime), ensuring irrationality of $\omega_k/2\pi$. By Kronecker’s and Weyl’s Equidistribution Theorems, this induces a dense, non-periodic, and ergodic trajectory: the process never exactly revisits any previous point, precluding repetitions in state and ensuring uniform coverage.

Because $\mathcal{R} \in SO(16)$, norm preservation holds ($\det \mathcal{R} = 1$), emphasizing unitarity. Empirically, numerical drift scales as $$E_{\mathrm{drift}}(t) \approx \sqrt{t}\,\delta_{\mathrm{machine}}$$, for float32 $\delta_{\mathrm{machine}} \sim 10^{-7}$, which is minor relative to the phonetic discrimination threshold even at $t \sim 10^{6}$.

2. Encoding, Evolution, and Retrieval: Mapping Language onto the Manifold

PTM operates by sequential application of three transformations per timestep $t$:

• Acoustic Injection $\Phi$: Each token $x_t$ is decomposed using IPA feature vectors and projected by a semi-orthogonal matrix $W_{\mathrm{proj}}$ to a 16-dimensional vector:

$$\Phi(x_t) = W_{\mathrm{proj}}[\text{IPA}(x_t)] \in \mathbb{R}^{16}$$

This is lossless for rhythm/phonetic content and specifically lossy for raw semantic content.

• Ergodic Evolution $\mathcal{R}$: The manifold state is updated recursively as

$$S_{t+1} = \mathcal{R} S_t \oplus \Phi(x_t) \pmod{1}$$

ensuring strictly unitary, non-decaying memory propagation.

• Manifold Resonance $D$: For retrieval, the process inverts the unitary dynamics to obtain the phonetic trace and computes cosine similarity with a vocabulary reference, combining this geometric evidence with the LLM prior for candidate selection.

This dynamical system can be summarized in the following algorithmic sketch:

def PTM_Encode(x_t): V_t = IPA_to_vector(x_t) # 16-dim phonetic injection S_t = (R @ S_{t-1} + V_t) % 1 # Unitary rotation + fold def PTM_Decode(S_t, S_{t-1}, C_size): V_rec = (S_t - R @ S_{t-1}) % 1 C = top_k_cosine(M_vocab, V_rec, k=C_size) for c in C: P_theta = ... # LLM prior P_phi = ... # Geometric likelihood return argmax_c [alpha * P_theta + (1-alpha) * P_phi]

3. Constant-Time Operations and Memory Compression

PTM memory is maintained as a single fixed-size vector $S_t \in \mathbb{T}^{16}$. Both encoding and decoding require:

• One $16\times16$ matrix multiply,
• One modular addition,
• A nearest-neighbor search in 16-D space,

All operations are invariant in time and space, i.e., retrieval and memory evolution are $O(1)$, independent of context length $t$.

Memory tokens are bifurcated into "Anchors" (high-entropy, preserved as sparse key-values) and "Bridges" (low-entropy, encoded solely in the manifold). For "Bridges," the compression is as follows:

• Standard dense KV: 4096 dimensions per token (8 KB, FP16).
• PTM: 16 dimensions, 64 bytes.

With an anchor drop rate of approximately 85–95% for Bridges, net compression exceeds $3,000\times$ compared to an end-to-end FP16 cache. For example, recalling 335 tokens from "Blind Walk" requires only 0.021 MB of PTM signals, compared to 64.32 MB for dense KV, reflecting $>3,000\times$ compression.

4. Signal Consensus and Hallucination Suppression

PTM retrieval deploys a dual-probability mechanism:

• Semantic Prior $P_{\theta}(c)$: Derived from the LLM, reflects standard predictive likelihood for candidates.
• Geometric Likelihood $P_{\phi}(c)$: Quantifies the proximity of a candidate phonetic vector to the unfolded manifold state:

$$P_{\phi}(c) = \text{softmax}_c(-\gamma \| (\mathcal{R}S_{t-1}\oplus\Phi(c)) - S_t \|_{\mathbb{T}})$$

• Consensus Mixture: The final selection blends these via

$$P_{\mathrm{total}}(c) = \alpha P_{\theta}(c) + (1 - \alpha) P_{\phi}(c), \quad \alpha \in [0, 1]$$

Empirically, $\alpha \approx 0.4$ yields strong suppression of hallucinated outputs and secures up to 92% accuracy on knowledge-centric tasks. The mechanism dynamically allocates trust: "Anchors" are rigid, while "Bridges" flex between semantic and geometric evidence according to confidence (Houichime et al., 23 Dec 2025).

5. Empirical Evaluation and Quantitative Performance

Experiments encompass datasets with varying entropy, including narrative text, scientific abstracts, and 20,000-token concatenations. Metrics include semantic accuracy (exact token recovery), compression ratio versus FP16 KV cache, and retrieval latency.

Dataset/Setting	Accuracy	Compression Ratio	Latency
20,000-token stream	89.2 ± 1.4%	4.4× (anchors + sig)	~14.1 µs (decode, CPU)
Sci-Fi narrative	92.34%	3.41×	≤35.6 ms (reconstruction, GPU)
Historical narrative	90.15%	3.64×
Blind walk (no anchors)	83.58%	>3,000×

Key findings include:

• Retrieval accuracy is invariant to token distance (no degradation with increased $t$).
• Memory requirements compress by orders of magnitude without degrading retrieval latency.
• Errors are primarily phonetic mutations, not semantic hallucinations.
• Proper noun anchors are recalled at 100% fidelity.
• Latency is negligible compared to LLM inference; worst-case reconstruction remains within interactive thresholds on modern hardware.

6. Limitations and Open Challenges

Several limitations and open issues remain:

• Anchor-Selection Irrevocability: Mislabeling of key tokens as low-entropy irreversibly embeds them in the manifold, precluding precise recovery.
• Narrative Domain Bias: In textual domains with low redundancy (code, cryptography), phonetic errors become critical, potentially catastrophic.
• Phonetic Homomorphism: Perfect homophones ("raise" vs. "raze") yield indeterminate retrieval; only semantic priors can disambiguate.
• Precision Constraints: High-stakes domains (legal, medical) are sensitive to minor phonetic corruption.
• Finite Precision Cycles: Although the cycle length of float32 arithmetic ($L_{\mathrm{sys}} \sim 2^{192}$) far exceeds practical limits, this is a theoretical, rather than operational, concern for human-scale text.

7. Contextual Significance and Theoretical Implications

The ergodic phonetic manifold concept reframes long-term memory in neural architectures from a collection of stored data to a continuous, dynamically evolving geometric process. By leveraging unitarity, ergodicity, and phonetic embedding, PTM demonstrates that infinite-context fidelity and constant-time access are practically attainable on finite hardware. This geometrization of memory, predicated on persistent acoustic traces and reconstructive resonance, suggests that scaling memory for LLMs need not be accompanied by proportional increases in hardware requirements. The approach opens avenues for further exploration in dynamical systems-based memory and raises new questions on optimal anchor selection, domain transfer, and theoretical bounds for context fidelity in symbolic and sub-symbolic sequential processes (Houichime et al., 23 Dec 2025).